Advanced Mathematics and Numerical Modeling of IoT

all messages passing through original MQTT message broker.
Consider

푆푏>푆푐=푟푎푝∗푟푝푑∗푡푑>푟푎푐∗푟푐푝∗푡푝+푟푐푝∗푟푝푑∗푡푑

(applying ( 6 ),( 11 )).

(12)

Note that the condition

푟푎푝>푟푐푝 (13)

should always satisfy to hold푆푏>푆푐,since푟푎푐∗푟푐푝∗푡푝is
always greater than zero. Consider

(푟푎푝−푟푐푝)∗푟푝푑∗푡푑>푟푎푐∗푟푐푝∗푡푝. (14)

In case that푟푎푝>푟푐푝,(14)is equivalent to

푟푝푑∗

푡푑

푡푝

>푟푎푐∗

푟푐푝

푟푎푝−푟푐푝

. (15)

Since푟푝푑=푡푝∗푚푝푑/푡푑(by(2)), we finally have the following:

푚푝푑>

푟푎푐∗푟푐푝

(푟푎푝−푟푐푝)

. (16)

To capture the meaning of conditions for푆푏 >푆푐,usinga
practical data, let us calculate the amount of total delivered
messages in SCondi and that of MQTT broker. We assume
that푡푑=100000,푡푝=80000,푡푎=5000,and푚푝푑=3.For
MQTTbrokercase,weassumetheaveragenumberofCPs
in a single application is 200. In SCondi, we assume that the
totalnumberofchannelsis2000,theaveragenumberofCPs
in a single channel is 50 and the average number of channels
in a single application is 4.Table 3shows the calculation of
the total delivered messages in both cases. As a result,푆푏
is 3000000 and푆푐is 1300000 when푚푝푑is 3. In this case,
both conditions(13)and(16)“푟푎푝(= 12.5) > 푟푐푝(= 1.25)”
and “푚푝푑(=3)>푟푎푝∗푟푐푝/(푟푎푝−푟푐푝)(=1.11)” are satisfied,
resulting in푆푐<푆푏.
Figure 5depicts the total amount of messages according
to the value of푚푝푑under the same conditions.Considering
the result, even in the aspect of the amount of total messages
passing through, the concept of context channel is very useful
and effective when CP is composed of two or more source
data.

6. Conclusion

In this paper, we introduced a context distribution frame-
work named SCondi that supports effective dissemination
of context data through context channels. SCondi is based
on two major components: channel selector and context
channel. The channel selector sends each raw context data
to each of the context channels that requires the contextual
data, using the MQTT messaging facility which has been
adopted by OASIS as a standard messaging facility for the
IoT. The context channel provides a filter chain mechanism
that supports effective extraction, tailoring, authentication,

Table 3: Example of delivered message comparison with practical

data.

Common conditions

푡푑=100000,푡푝=80000,푡푎=5000,푚푝푑=3

MQTT broker case SCondi case

Let

푚푎푝=200

Let

푡푐=2000,푚푎푐=4,푚푐푝=50

푆푏 =푡푎∗푚푎푝∗푚푝푑

=5000∗200∗3

=3000000

푆푐 =푡푎∗푚푎푐∗푚푐푝

+푡푐∗푚푐푝∗푚푝푑

=5000∗4∗50

+2000∗50∗3

=1300000

푟푝푑 =푡푝∗푚푝푑/푡푑

=80000∗3/100000

=2.4

푟푐푝 =푡푐∗푚푐푝/푡푝

=2000∗50/80000

=1.25

푟푎푝 =푡푎∗푚푎푝/푡푝

=5000∗200/80000

=12.5

푟푎푐 =푡푎∗푚푎푐/푡푐

=5000∗4/2000

=10

푚푝푑 >푟푎푐∗푟푐푝/(푟푎푝−푟푐푝)

>10∗1.25/(12.5−1.25)

>1.11

0

1

2

3

4

5

6

7

8

1234567

Sb: total messages through MQTT broker

Sc: total messages through SCondi

mpd

×10^6

Number of messages

Figure 5:푚푝푑effect on total amount of messages.

and security of information through various types of filters.

Based on the MQTT messaging facility again, when receiving

the collection of the associated contextual data, the context

channel delivers the collection of data to each subscriber

of the channel after processing the collection of data with

the filter chain. In addition, SCondi supports three types of